340 Consistency and Limiting Distributions
5.2.10.Modify the R functioncdistpltto show histograms of the valueswn
discussed in Example 5.2.8.
5.2.11.Letp=0.95 be the probability that a man, in a certain age group, lives at
least 5 years.
(a)If we are to observe 60 such men and if we assume independence, use R to
compute the probability that at least 56 of them live 5 or more years.
(b)Find an approximation to the result of part (a) by using the Poisson distri-
bution.
Hint:Redefinepto be 0.05 and 1−p=0.95.
5.2.12.Let the random variableZnhave a Poisson distribution with parameter
μ=n. Show that the limiting distribution of the random variableYn=(Zn−n)/
√
n
is normal with mean zero and variance 1.
5.2.13.Prove Theorem 5.2.3.
5.2.14.LetXnandYnhave a bivariate normal distribution with parametersμ 1 ,μ 2 ,
σ 12 ,σ 22 (free ofn) butρ=1− 1 /n. Consider the conditional distribution ofYn,given
Xn=x. Investigate the limit of this conditional distribution asn→∞.Whatis
the limiting distribution ifρ=−1+1/n? Reference to these facts is made in the
remark of Section 2.5.
5.2.15. LetXndenote the mean of a random sample of sizenfrom a Poisson
distribution with parameterμ=1.
(a)Show that the mgf of Yn =
√
n(Xn−μ)/σ =
√
n(Xn−1) is given by
exp[−t
√
n+n(et/
√n
−1)].
(b)Investigate the limiting distribution ofYnasn→∞.
Hint:Replace, by its MacLaurin’s series, the expressionet/
√n
, which is in the
exponent of the mgf ofYn.
5.2.16.Using Exercise 5.2.15 and the Δ-method, find the limiting distribution of
√
n(
√
Xn−1).
5.2.17.LetXndenote the mean of a random sample of sizenfrom a distribution
that has pdff(x)=e−x, 0 <x<∞, zero elsewhere.
(a)Show that the mgfMYn(t)ofYn=
√
n(Xn−1) is
MYn(t)=[et/
√n
−(t/
√
n)et/
√n
]−n,t<
√
n.
(b)Find the limiting distribution ofYnasn→∞.
Exercises 5.2.15 and 5.2.17 are special instances of an important theorem that will
be proved in the next section.
5.2.18.Continuing with Exercise 5.2.17, use the Δ-method to find the limiting
distribution of
√
n(
√
Xn−1).